The quater-imaginary numeral system is a numeral system, first proposed by Donald Knuth in 1960. Unlike standard numeral systems, which use an integer (such as 10 in decimal, or 2 in binary) as their bases, it uses the imaginary number ${\displaystyle 2i}$ (such that ${\displaystyle (2i)^{2}=-4}$) as its base. It is able to (almost) uniquely represent every complex number using only the digits 0, 1, 2, and 3. Numbers less than zero, which are ordinarily represented with a minus sign, are representable as digit strings in quater-imaginary; for example, the number −1 is represented as "103" in quater-imaginary notation.

In a positional system with base ${\displaystyle b}$,

${\displaystyle \ldots d_{3}d_{2}d_{1}d_{0}.d_{-1}d_{-2}d_{-3}\ldots }$

represents${\displaystyle \dots +d_{3}\cdot b^{3}+d_{2}\cdot b^{2}+d_{1}\cdot b+d_{0}+d_{-1}\cdot b^{-1}+d_{-2}\cdot b^{-2}+d_{-3}\cdot b^{-3}\dots }$

In this numeral system, ${\displaystyle b=2i}$,

and because ${\displaystyle (2i)^{2}=-4}$, the entire series of powers can be separated into two different series, so that it simplifies to ${\displaystyle {\begin{aligned}&{}[\dots +d_{4}\cdot (-4)^{2}+d_{2}\cdot (-4)^{1}+d_{0}+d_{-2}\cdot (-4)^{-1}+\dots ]\end{aligned}}}$ for even-numbered digits (digits that simplify to the value of the digit times a power of -4), and ${\displaystyle {\begin{aligned}2i\cdot [\dots +d_{3}\cdot (-4)^{1}+d_{1}+d_{-1}\cdot (-4)^{-1}+d_{-3}\cdot (-4)^{-2}+\dots ]\end{aligned}}}$ for those digits that still have an imaginary factor. Adding these two series together then gives the total value of the number.

Because of the separation of these two series, the real and imaginary parts of complex numbers are readily expressed in base −4 as ${\displaystyle \ldots d_{4}d_{2}d_{0}.d_{-2}\ldots }$ and ${\displaystyle 2\cdot (\ldots d_{5}d_{3}d_{1}.d_{-1}d_{-3}\ldots )}$ respectively.

To convert a digit string from the quater-imaginary system to the decimal system, the standard formula for positional number systems can be used. This says that a digit string ${\displaystyle \ldots d_{3}d_{2}d_{1}d_{0}}$ in base b can be converted to a decimal number using the formula